Archer is the Admin of the infamous group known as
Team Rocket, the true antagonists of the game and the same you will have to defeat on a number of occasions during the Main Storyline. Even though you will encounter Team Rocker on a number of occasions during your journey through the Kanto Region, you will actually only battle Archer on three of those instances (one of which won't be until after defeating the
Elite Four and entering the Post Game).
You will first meet Archer at the very end of the
Rocket Hideout, directly after
Team Rocket killed a Marowak back in
Lavender Town. Archer will be discussing with
Giovanni about the technology found at
Silph Co in
Saffron City, and Giovanni will give him the
Green light to take over said company. Archer will leave quickly after this, but will notice you as he does. Upset about how you have infiltrated the hideout, as well as how you have been eavesdropping on the conversation he had with Giovanni, Archer will challenge you to a battle. He will use Weezing and Golbat when in battle, both at Level 33.
The next time you encounter Archer will be on the fifth floor of
Silph Co while
Team Rocket has taken over the building in
Saffron City. He will be in the same room as the Team Rocket Grunt that has the Card Key, the same that will grant you access to every room in Silph Co. You will then be challenged by both Archer and this Grunt to a battle, with your
Rival teaming up with you in order to stop them. Archer will use Electrode, Weezing and Golbat in this Multi-Battle, all of which will be at Level 37. By defeating both Archer and this other Team Rocket Member in battle with your Rival, you will ultimately obtain the Card Key.
You will not see Archer again until after you have become the newest Kanto Region
Champion. He will be found back in the
Rocket Hideout, and he will question what your trainer is doing there since
Team Rocket disbanded. Before long, he will challenge you to a battle. All of his Pokemon will be at Level 54 this time around, and he will use none other than Electrode, Weezing, Golbat and Magmar. After defeating Archer, he will read you a letter he actually wrote you. He will say that he holds no grudges against your trainer, but he will make mention of him leaving the Kanto Region in order to build an even stronger Team Rocket. Archer will tear the letter before leaving once and for all.